Chapter 01

Chapter 01 - Introduction The time-until-death random...

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FIN 3220A Actuarial Models I Chapter 1 Survival Distributions and Life Tables FIN 3220A (2005-2006) Chapter 1 Survival Distributions and Life Tables Page 2 Introduction ± The time-until-death random variable, T ( x ), is the basic building block in life insurance and annuity ± Another important distribution is the distribution of the age-at-death random variable X ± A life table can be used to summarize a distribution of the age-at-death random variable ± Life tables are used to build models for insurance systems designed to assist individuals facing uncertainty about the times of their deaths FIN 3220A (2005-2006) Chapter 1 Survival Distributions and Life Tables Page 3 The Survival Function ± Consider a newborn child ± This newborn’s age-at-death, X , is a continuous type random variable ± We denote the distribution function (d.f.) of X as F X ( x ) ± We define the survival function (s.f.), s ( x ), as ± As we always assume that F X (0) = 0, and hence s (0) = 1 ± For any positive x , s ( x ) is the probability of a newborn will attain age x ± We can deduce the properties of s.f. from d.f., or vice versa ± We can make probability statements about the age-to-death in terms of either s.f. or d.f., for example, with 0 < x < z () ( ) Pr 0 X Fx Xx x =≤ () () ( ) 1P r 0 X sx F x X x x =− = > () ( ) ( ) () () Pr XX xXz Fz Fx sx sz <≤= FIN 3220A (2005-2006) Chapter 1 Survival Distributions and Life Tables Page 4 Time-until-Death for a Person Age x ± The conditional probability that a newborn will die between the ages x and z , given survival to age x , is ± We usually denote a life-age-x by the symbol ( x ) ± The future lifetime of ( x ), X x , is denoted by T ( x ) ± A standard set of actuarial symbols are used to promote research and communication ² The standard is called the International Actuarial Notation (IAN) ² These symbols differ from those used for probability notation Pr and Pr Pr Pr Pr 1 X xXz X xX x Fz Fx sx s z s x <≤ > > = = >> −− ==
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FIN 3220A (2005-2006) Chapter 1 Survival Distributions and Life Tables Page 5 Time-until-Death for a Person Age x ± To make probability statements about T ( x ), we use the notations ± The symbol t q x is the probability that ( x ) will die within t years ² That is, t q x is the d.f. of T ( x ) ± The symbol t p x is the probability that ( x ) will attain age x + t ² That is, t p x is the s.f. for ( x ) ± For a special case of a life-age-0, we have T (0) = X and () [] Pr 0, Pr 1 0 tx qT x t t pT x tq t =≤ => = 0 0 t ps t t =≥ FIN 3220A (2005-2006) Chapter 1 Survival Distributions and Life Tables Page 6 Time-until-Death for a Person Age x ± If t = 1, convention permits us to omit the prefix in the symbols, we have ± Special symbol is used for the more general event that ( x ) will survive t years and die within the following u years ² That is, ( x ) will die between ages x + t and x + t + u ± As before, if u = 1, the prefix is deleted in , and we have Pr will die within 1 year , Pr will attain age 1 x x qx px x = =+ Pr Pr Pr x tu tu x t x tx t ux qt T x t u Tx t u Tx t qq pp + + =< + + −≤ =− x q x t q FIN 3220A (2005-2006) Chapter 1 Survival Distributions and Life Tables Page 7 Time-until-Death for a Person Age x ±
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Chapter 01 - Introduction The time-until-death random...

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